In most incarnations, Brainac utilizes the skull ship as his mobile base in which he cruises the universe shrinking and bottling cities from a myriad of planets.
Pre-Crisis ship was based of the design of Brainiac himself, at that time a menacing robot sporting a head of the same shape. This ship was last seen in Crisis on Infinite Earths, where the villains used it in their own efforts to thwart the Anti-Monitor.
The Skull Ship is also the site from where Darkseid launched his decisive attack on the Anti-Monitor, leading to the end of the cosmic villain.
After Crisis on Infinite Earth the Skull Ship was written out of continuity as Brainiac was reinvented and his city-stealing ways forgotten about. The ship was reintroduced in Action Comics: Brainiac storyline, which brought the old Pre-Crisis Brainiac into continuity.
Once again the ship is Brainiac's mobile base and repository of bottled cities. The ship is also extremely formidable, with shields capable of protecting it against even supernova explosions and it is able to attack and preform its city-shrinking function with impunity regardless of the force mobilized against it. It also boasts an army of robotic drones who perform Brainiac's bidding, whether it is scientific experiments, slaughter of resisting troupes or just manual labor. These drones sport the same style as the Skull Ship itself, and are in fact, based on Brainiac's Pre-Crisis design. It also had special weapons designed to destroy suns.
With his ship, Brainiac was has abducted dozens of cities from all over the universe. Each time he took a city, he turned the planet's sun into a supernova, destroying the civilization and making himself the sole possessor of it's knowledge. In the past he has visited Krypton and shrank the city of Kandor, Krypton's army being powerless to stop him.
He was prevented, however, to destroy their sun and, in turn, Krypton itself. He later returned to Krypton's surviving colony, Argo City, accidentally called by arogue left behind drone. The survivors were integrated into the Bottle City of Kandor.
Superman first made contact with the Skull Ship when he went on an alien planet to confront Brainiac. He was unsuccessful, and Brainiac casually destroyed the planet's sun, making his way to Earth next.
On Earth, Superman was capable of defeating Brainiac. The bottled city of Kandor was restored and the Skull Ship (with Brainiac himself) was taken into US Army custody. There it was examined by scientists under the command of General Sam Lane, looking for ways to create weapons against Kryptonians.
Eventually Lex Luthor, then also a prisoner of Lane's, managed to free Brainiac and helped him take control of his ship. Together they annihilated the US Army base and made way to New Krypton, Brainiac intending to reclaim what was taken from him: Kandor.
The Skull Ship attacked Kandor, and soon a three way battle between Brainac's drones (enhanced by kryptonite and red sun emitters), the Kryptonians and the Legion of Superheroes (there to recover Brainiac's bottled cities).
In the end, the Skull Ship was destroyed when Lex broke the containment field (bottle) of one of the cities, re-enlarging it with the ship, causing the ship to explode.
The New 52
Although this time the ship lacked the shape of a skull (as it originally appeared Pre-Crisis, but Post-Crisis as well), Brainiac's ship made its appearance in the new DCU in the pages of Action Comics (in the final page of issue #2). Once again serving its purpose as a vessel for shrunken cities and other significant artifacts stolen from various alien worlds, it was Brainiac's base of operations. In the interlude stories of Action Comics #5 & #6 it was revealed that after Brainiac's defeat (by Superman), the ship served as Superman's original Fortress of Solitude.